December 5, 2025Open Access

Structure and Computation

Key Points

Isomorphic number systems can differ significantly in computability-theoretic properties, undermining a core assumption.
The study shows that these properties are crucial for practical arithmetic, challenging structuralism's relevance claims.
Analysis confronts arithmetical structuralists with a dilemma—irrelevant properties become relevant for arithmetic.
The findings indicate that arithmetical structuralism may not adequately reflect the nature of arithmetic frameworks.

Abstract

ABSTRACT It is a truism of mathematics that differences between isomorphic number systems are irrelevant to arithmetic. This truism is deeply rooted in the modern axiomatic method and underlies most strands of arithmetical structuralism, the view that arithmetic is about some abstract number structure. In this paper, I challenge this truism by showing that isomorphic systems can differ with regard to important computational features of numbers. This confronts arithmetical structuralists with a dilemma. On the one hand, many computability‐theoretic properties are only satisfied by particular number systems, and are hence disqualified as irrelevant by structuralist accounts. On the other hand, these properties turn out to be highly relevant to arithmetical practice. Hence, as I argue, arithmetical structuralism is not a tenable view about arithmetic.

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